I've searched the net for examples of how to use residue to solve an inverse Laplace transform when you have double poles but so far I've found nothing good. Every time I try to do it on my own I end up losing! So here's my current problem:
$$
L^{-1}\left(\frac{1}{(s^2+1)^2}\right)
$$
This should be solvable using the rule
$$
\mathrm{Res}(f,c) = \frac{1}{(n-1)!} \lim_{z \to c} \frac{d^{n-1}}{dz^{n-1}}\left( (z-c)^{n}f(z) \right)
$$
But I can't seem to get it right when setting $g_1(s)=e^{st}/(s-i)^2$ and $g_2(s)=e^{st}/(s+i)^2$ then using the rule above to get
$$
f(t)=g_1'(i) + g_1'(-i)
$$
Ideas?